For a 0.200 m downward displacement of the 8.00 kg block, what is the change in the gravitational potential energy associated with each block? Express your answers in joules separa... For a 0.200 m downward displacement of the 8.00 kg block, what is the change in the gravitational potential energy associated with each block? Express your answers in joules separated by a comma.
Understand the Problem
The question asks for the change in gravitational potential energy associated with two blocks of different masses when one block is displaced downward by 0.200 m. It asks for these changes to be expressed in joules.
Answer
The change in gravitational potential energy for the blocks is $-15.696 \, \text{J}, 11.772 \, \text{J}$.
Answer for screen readers
The change in gravitational potential energy for the blocks is: $$ -15.696 , \text{J}, 11.772 , \text{J} $$
Steps to Solve
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Identify the known values We have the mass of the first block, ( m_1 = 8.00 , \text{kg} ) and the mass of the second block, ( m_2 = 6.00 , \text{kg} ). The downward displacement of the first block is ( d = 0.200 , \text{m} ).
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Understand gravitational potential energy change The change in gravitational potential energy (( \Delta PE )) of an object is given by the formula: $$ \Delta PE = m \cdot g \cdot h $$ where:
- ( m ) is the mass of the object
- ( g ) is the acceleration due to gravity (approximately ( 9.81 , \text{m/s}^2 ))
- ( h ) is the change in height (displacement)
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Calculate the change in potential energy for the first block The 8.00 kg block moves down, so the change in height is negative: $$ \Delta PE_1 = m_1 \cdot g \cdot (-d) $$ Substituting the values: $$ \Delta PE_1 = 8.00 , \text{kg} \cdot 9.81 , \text{m/s}^2 \cdot (-0.200 , \text{m}) $$
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Calculate the change in potential energy for the second block The 6.00 kg block moves up, so the change in height is positive: $$ \Delta PE_2 = m_2 \cdot g \cdot d $$ Substituting the values: $$ \Delta PE_2 = 6.00 , \text{kg} \cdot 9.81 , \text{m/s}^2 \cdot 0.200 , \text{m} $$
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Calculate final values Now compute ( \Delta PE_1 ) and ( \Delta PE_2 ):
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For ( \Delta PE_1 ): $$ \Delta PE_1 = 8.00 \cdot 9.81 \cdot (-0.200) = -15.696 , \text{J} $$
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For ( \Delta PE_2 ): $$ \Delta PE_2 = 6.00 \cdot 9.81 \cdot 0.200 = 11.772 , \text{J} $$
The change in gravitational potential energy for the blocks is: $$ -15.696 , \text{J}, 11.772 , \text{J} $$
More Information
The changes in gravitational potential energy occur due to the displacement of the blocks. The 8.00 kg block loses potential energy as it moves downward, while the 6.00 kg block gains potential energy as it moves upward.
Tips
- Confusing the direction of movement with the sign of potential energy change; remember that downward displacement results in negative potential energy.
- Using incorrect values for gravitational acceleration; always use ( g \approx 9.81 , \text{m/s}^2 ).
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